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3   THE MODULUS FUNCTION ON A GROUP       259

with a matrix for which £.,„..* = 0 if r &j, and £Jtn_k = 1. Finally, multiply-
ing this matrix by £jtn_k(- l)Bn-ktJ(l), we obtain a matrix of the same form
but with j = n — k. This completes the induction step, and hence also the
proof of the lemma.

Bearing in mind (14.3.8) we see that in order to prove (14.3.9) it is enough
to consider the cases where u takes one of the forms

(14.3.9.2)              (XL . . . , *„)*->(*!, . . . , *„__!, axj,

(14.3.9.3)              (XL ...,xj\-*(xi, ...,xyH- Axt.,... ,*:„).

Now by (13.21.2), if /is any function belonging to Jf(R"), we have

f   r

• • •     f(xl9 . . . , Xn_L #*„) </Xi  • ' ' rfxw

J        J

r   f                   r

=     • • • \dx± - • • <£*„_!   /(xl9 . . . , 0xn) dxn

J       J                               J

and the formula for change of variable (8.7.4) gives

p-foo                                                                         /* + oo

(X15 ...,^_1,^n)^= M'1 1          f(xl9 ...,Xn)dxn,

J~oo                                                       J-oo

so that (14.3.9) is true when u is of the form (14.3.9.2). Likewise, we may write

r   r

• • •    /(*!, . . . , xj + Axf , . . . , xn) dxl dx2 - • • ^

J      J

=    • • • \dxl - - dxj^l dxj+t '"dxn    f(xl9 . . . , Xj -f ijct, . . ., xn) dxy,

and by virtue of the fact that Lebesgue measure is translation-invariant,

r+oo                                                              r+«>

f(xl9 . . . , xj + tei , . . . , xn) dxj =        /(x, , ...9xj9 ...9xn)dXj.

J-oo                                                                J~oo

This proves (14.3.9) when w is of the form (14.3.9.3).

(14.3.10)   Application to the calculation of integrals:   I. Parallelotope and
simplex.

In R", let P be the " parallelotope " constructed on n linearly independent

n

vectors xit . . . , xn, i.e., the set of all vectors £ f 4xf with 0 <£ ^- S 1 for eachk. (Show that Zp is the union of pk closed balls of radius